Graded identities of matrix algebras and the universal graded algebra

We consider fine group gradings on the algebra $M_n(\mathbb{C})$ of $n$ by $n$ matrices over the complex numbers and the corresponding graded polynomial identities. Given a group $G$ and a fine $G$-grading on $M_n(\mathbb{C})$, we show that the $T$-ideal of graded identities is generated by a special type of identity, and, as a consequence, we solve the corresponding Specht problem for this case. Next we construct a universal algebra $U$ (depending on the group $G$ and the grading) in two different ways: one by means of polynomial identities and the other one by means of a generic two-cocycle (this parallels the classical constructions in the nongraded case). We show that a suitable central localization of $U$ is Azumaya over its center and moreover, its homomorphic images are precisely the $G$-graded forms of $M_n(\mathbb{C})$. Finally, we consider the ring of central quotients of $U$ which is a central simple algebra over the field of quotients of the center of $U$. Using earlier results of the authors we show that this is a division algebra if and only if the group $G$ is one of a very explicit (and short) list of nilpotent groups. It follows that for groups not on this list, one can find a nonidentity graded polynomial such that its power is a graded identity. We illustrate this phenomenon with an explicit example.